Odd Function

Q. If $f:[-2,2]\to R$ defined by $f\left(x\right)=x^3+\tan x+\left[\frac{x^2+1}{p}\right]$ is an odd function, then the least value of $\left[p\right]$ is
($[.]$ represents the greatest integer function)

Q. Let $f\left(x\right)=\{\begin{array}{cc}{x}^2-3,& x\le -5\\ x+\lambda ,& -5<x<-1\\ (\mu -7)(|1-x|+|1+x|),& -1\le x\le 1\\ x+6,& 1<x<5\\ 3-x^2,& x\ge 5\end{array}$
If $f\left(x\right)$ is an odd function, then the value of $\lambda +\mu$ is

Q. If $f\left(x\right)=\frac{\cos x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}},$ where $x$ is not an integral multiple of $\pi$ and $[.]$ denotes the greatest integer function, then

1. $f\left(x\right)$ is an even function
2. $f\left(x\right)$ is an odd function
3. $f\left(x\right)$ is neither even nor odd
4. None of these

Q. $\text{List I}$ contains functions and $\text{List II}$ contains behaviours of functions in their respective domain. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& f\left(x\right)={\left(\right(\text{sgn}x{)}^{\text{sgn}x})}^n;x\ne 0& \text{(P)}& \text{odd function}\\ & \text{where}n\text{is an odd integer and}& & \\ & \text{sgn}x\text{denotes the signum function}& & \\ \text{(B)}& f\left(x\right)=\frac{x}{e^x-1}+\frac{x}{2}+1& \text{(Q)}& \text{even function}\\ \text{(C)}& f\left(x\right)=\{\begin{array}{cc}0,& \text{if}x\text{is rational}\\ 1,& \text{if}x\text{is irrational}\end{array}& \text{(R)}& \text{neither odd nor}\\ & & & \text{even function}\\ \text{(D)}& f\left(x\right)=max\{\tan x,\cot x\}& \text{(S)}& \text{periodic}\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(P)},\text{(S)}$
2. $\text{(B)}\to \text{(Q)}$
3. $\text{(A)}\to \text{(Q)},\text{(S)}$
4. $\text{(B)}\to \text{(R)}$

Q. $\text{List I}$ contains functions and $\text{List II}$ contains behaviours of functions in their respective domain. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& f\left(x\right)={\left(\right(\text{sgn}x{)}^{\text{sgn}x})}^n;x\ne 0& \text{(P)}& \text{odd function}\\ & \text{where}n\text{is an odd integer and}& & \\ & \text{sgn}x\text{denotes the signum function}& & \\ \text{(B)}& f\left(x\right)=\frac{x}{e^x-1}+\frac{x}{2}+1& \text{(Q)}& \text{even function}\\ \text{(C)}& f\left(x\right)=\{\begin{array}{cc}0,& \text{if}x\text{is rational}\\ 1,& \text{if}x\text{is irrational}\end{array}& \text{(R)}& \text{neither odd nor}\\ & & & \text{even function}\\ \text{(D)}& f\left(x\right)=max\{\tan x,\cot x\}& \text{(S)}& \text{periodic}\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(R)}$
2. $\text{(D)}\to \text{(P)},\text{(S)}$
3. $\text{(C)}\to \text{(Q)}$
4. $\text{(D)}\to \text{(R)},\text{(S)}$